Capacitance and charge stored in a parallel-plate capacitor
(a) What is the capacitance of an empty parallel-plate capacitor with metal plates that each have an area of
, separated by 1.00 mm? (b) How much charge is stored in this capacitor if a voltage of
is applied to it?
Strategy
Finding the capacitance
C is a straightforward application of
[link] . Once we find
C , we can find the charge stored by using
[link] .
This small capacitance value indicates how difficult it is to make a device with a large capacitance.
Inverting
[link] and entering the known values into this equation gives
Significance
This charge is only slightly greater than those found in typical static electricity applications. Since air breaks down (becomes conductive) at an electrical field strength of about 3.0 MV/m, no more charge can be stored on this capacitor by increasing the voltage.
Suppose you wish to construct a parallel-plate capacitor with a capacitance of 1.0 F. What area must you use for each plate if the plates are separated by 1.0 mm?
Each square plate would have to be 10 km across. It used to be a common prank to ask a student to go to the laboratory stockroom and request a 1-F parallel-plate capacitor, until stockroom attendants got tired of the joke.
A spherical capacitor is another set of conductors whose capacitance can be easily determined (
[link] ). It consists of two concentric conducting spherical shells of radii
(inner shell) and
(outer shell). The shells are given equal and opposite charges
and
, respectively. From symmetry, the electrical field between the shells is directed radially outward. We can obtain the magnitude of the field by applying Gauss’s law over a spherical Gaussian surface of radius
r concentric with the shells. The enclosed charge is
; therefore we have
Thus, the electrical field between the conductors is
We substitute this
into
[link] and integrate along a radial path between the shells:
In this equation, the potential difference between the plates is
. We substitute this result into
[link] to find the capacitance of a spherical capacitor:
Capacitance of an isolated sphere
Calculate the capacitance of a single isolated conducting sphere of radius
and compare it with
[link] in the limit as
.
Strategy
We assume that the charge on the sphere is
Q , and so we follow the four steps outlined earlier. We also assume the other conductor to be a concentric hollow sphere of infinite radius.
Solution
On the outside of an isolated conducting sphere, the electrical field is given by
[link] . The magnitude of the potential difference between the surface of an isolated sphere and infinity is
The capacitance of an isolated sphere is therefore
Significance
The same result can be obtained by taking the limit of
[link] as
. A single isolated sphere is therefore equivalent to a spherical capacitor whose outer shell has an infinitely large radius.